Cutup - Futility Closet

Cutup

A problem by Russian mathematician Viktor Prasolov: Prove that it’s impossible to cut a 10×10 chessboard into T-shaped tiles of 4 squares each.

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Suppose that the task is possible, and that we’ve done it. Due to their shape, each of our 25 tiles contains either 1 or 3 black chessboard squares, always an odd number. So, taken together, the tiles contain an odd number of black squares. But the original board contained 100/2 = 50 black squares. That’s a contradiction, so the task is not possible. (From Prasolov’s Problems in Plane and Solid Geometry.)